Question

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Part 1
Write a​ slope-intercept equation for a line passing through the point left parenthesis 4 comma negative 2 right parenthesis(4,−2) that is parallel to the line 4 x plus 5 y equals 74x+5y=7. Then write a second equation for a line passing through the point left parenthesis 4 comma negative 2 right parenthesis(4,−2) that is perpendicular to the line 4 x plus 5 y equals 74x+5y=7.
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Part 1
The equation of the parallel line is enter your response here.

Answer 1

To find the equations of lines that are parallel and perpendicular to a given line, we first need to determine the slope of the given line.

### Step 1: Find the Slope of the Given Line
The original equation is:

\[
4x + 5y = 7
\]

To find the slope, we can rearrange this into slope-intercept form \(y = mx + b\), where \(m\) is the slope.

1. Subtract \(4x\) from both sides:
\[
5y = -4x + 7
\]
2. Divide everything by \(5\):
\[
y = -\frac{4}{5}x + \frac{7}{5}
\]

From this, we see that the slope \(m\) of the line is \(-\frac{4}{5}\).

### Step 2: Equation of the Parallel Line
Since parallel lines have the same slope, the slope of the line we want to find will also be \(-\frac{4}{5}\). We will use the point \((4, -2)\) and the point-slope form of the equation of a line:

\[
y - y_1 = m(x - x_1)
\]

Using \((x_1, y_1) = (4, -2)\) and \(m = -\frac{4}{5}\):

\[
y - (-2) = -\frac{4}{5}(x - 4)
\]

Now, simplify this equation:

\[
y + 2 = -\frac{4}{5}x + \frac{16}{5}
\]

Subtract 2 from both sides (which is the same as \(-\frac{10}{5}\)):

\[
y = -\frac{4}{5}x + \frac{16}{5} - \frac{10}{5}
\]
\[
y = -\frac{4}{5}x + \frac{6}{5}
\]

Thus, the equation of the parallel line is:

\[
\boxed{y = -\frac{4}{5}x + \frac{6}{5}}
\]

### Step 3: Equation of the Perpendicular Line
To find the slope of the line that is perpendicular to the original line, we take the negative reciprocal of the slope \(-\frac{4}{5}\). The negative reciprocal is:

\[
\frac{5}{4}
\]

Now we will use the point \( (4, -2) \) again and this new slope:

Using \(m = \frac{5}{4}\):

\[
y - (-2) = \frac{5}{4}(x - 4)
\]

Simplifying:

\[
y + 2 = \frac{5}{4}x - 5
\]

Subtract \(2\) (which is the same as \(\frac{8}{4}\)) from both sides:

\[
y = \frac{5}{4}x - 5 - 2
\]
\[
y = \frac{5}{4}x - 7
\]

Thus, the equation of the perpendicular line is:

\[
\boxed{y = \frac{5}{4}x - 7}
\]

In summary:

1. The equation of the parallel line is \(y = -\frac{4}{5}x + \frac{6}{5}\).
2. The equation of the perpendicular line is \(y = \frac{5}{4}x - 7\).